A non-diagonalizable pure state

Abstract

We construct a pure state on the C*-algebra B(ℓ2) of all bounded linear operators on ℓ2, which is not diagonalizable [i.e., it is not of the form limu⟨T(ek),ek⟩ for any orthonormal basis (ek)k∈N of ℓ2 and an ultrafilter u on N]. This constitutes a counterexample to Anderson’s conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah, and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison–Singer problem due to A. Marcus, D. Spielman, and N. Srivastava that the restriction of our pure state to any atomic masa D((ek)k∈N) of diagonal operators with respect to an orthonormal basis (ek)k∈N is not multiplicative on D((ek)k∈N).